This article provides a comprehensive overview of continuous-time quantum Monte Carlo (CT-QMC) methods, which are advanced techniques for solving quantum impurity models. Quantum impurity models describe an atom or molecule embedded in a host material, and they are crucial in nanoscience and the study of correlated electron materials. The article discusses the limitations of previous methods, such as the Hirsch-Fye quantum Monte Carlo method, and highlights the advantages of CT-QMC methods, which avoid explicit time discretization and address the sign problem more effectively. The authors present detailed derivations and descriptions of several CT-QMC algorithms, including the diagrammatic Monte Carlo method, the interaction expansion algorithm (CT-INT), the auxiliary field algorithm (CT-AUX), and the hybridization expansion solver (CT-HYB). They also introduce the continuous-time Jreservation (CT-J) method for Kondo-like problems. Each algorithm is explained in sufficient detail to allow readers to implement them independently. The article reviews the strengths and weaknesses of CT-QMC methods, their successful applications to various problems, and future prospects. It emphasizes the broad applicability of these methods to a wide range of physically realistic models, including those with high and low energy scales. The review also discusses the challenges and limitations, such as the sign problem in fermionic models, and the need for further development to fully address these issues. Overall, the article aims to provide a thorough understanding of CT-QMC methods and their potential to advance research in nanoscience, correlated electron physics, and nonequilibrium systems.This article provides a comprehensive overview of continuous-time quantum Monte Carlo (CT-QMC) methods, which are advanced techniques for solving quantum impurity models. Quantum impurity models describe an atom or molecule embedded in a host material, and they are crucial in nanoscience and the study of correlated electron materials. The article discusses the limitations of previous methods, such as the Hirsch-Fye quantum Monte Carlo method, and highlights the advantages of CT-QMC methods, which avoid explicit time discretization and address the sign problem more effectively. The authors present detailed derivations and descriptions of several CT-QMC algorithms, including the diagrammatic Monte Carlo method, the interaction expansion algorithm (CT-INT), the auxiliary field algorithm (CT-AUX), and the hybridization expansion solver (CT-HYB). They also introduce the continuous-time Jreservation (CT-J) method for Kondo-like problems. Each algorithm is explained in sufficient detail to allow readers to implement them independently. The article reviews the strengths and weaknesses of CT-QMC methods, their successful applications to various problems, and future prospects. It emphasizes the broad applicability of these methods to a wide range of physically realistic models, including those with high and low energy scales. The review also discusses the challenges and limitations, such as the sign problem in fermionic models, and the need for further development to fully address these issues. Overall, the article aims to provide a thorough understanding of CT-QMC methods and their potential to advance research in nanoscience, correlated electron physics, and nonequilibrium systems.